Question

the graph of the function $g(x) = -|x|$ is shown on the grid below.
graph the function $h(x) = -|x + 2| + 3$ in the interactive graph.

Explanation:

Step1: Find the vertex

The vertex form of an absolute value function is \(a|x - h| + k\), where \((h, k)\) is the vertex. For \(h(x) = -|x + 2| + 3\), rewrite \(x + 2\) as \(x - (-2)\), so vertex is \((-2, 3)\).

Step2: Determine the slope

The coefficient of the absolute value term is \(-1\), so the graph opens downward with slopes of \(1\) (left of vertex) and \(-1\) (right of vertex).

Step3: Plot key points

From the vertex \((-2, 3)\):

• Left: Move 1 left (\(x=-3\)), \(y=3 - 1=2\) → \((-3,2)\); move 2 left (\(x=-4\)), \(y=3 - 2=1\) → \((-4,1)\).
• Right: Move 1 right (\(x=-1\)), \(y=3 - 1=2\) → \((-1,2)\); move 2 right (\(x=0\)), \(y=3 - 2=1\) → \((0,1)\).

Step4: Draw the graph

Connect these points to form the V-shaped graph opening downward with vertex at \((-2,3)\).

Answer:

The vertex of \(h(x) = -|x + 2| + 3\) is at \((-2, 3)\). The graph opens downward with a slope of \(\pm1\). Key points: \((-2, 3)\), \((-3, 2)\), \((-1, 2)\), \((-4, 1)\), \((0, 1)\).